jats:pIn this article, we extend Eardley and Moncrief’s L∞ estimates [Commun. Math. Phys. 83(2), 193–212 (1982)] for the conformally invariant Yang–Mills–Higgs equations to the Einstein cylinder. Our method is to first work on Minkowski space and localize their estimates and then carry them to the Einstein cylinder by a conformal transformation. By patching local estimates together, we deduce global L∞ estimates on the cylinder and extend Choquet-Bruhat and Christodoulou’s [Ann. Sci. Éc. Norm. Supér. 14(4), 481–506 (1981)] small data well-posedness result to large data. Finally, by employing an inverse conformal transformation, we deduce exponential decay rates for Yang–Mills–Higgs fields on de Sitter space and inverse polynomial decay rates on Minkowski space.</jats:p>